Global Solution for the Optimal Feedback Control of the Underactuated Heisenberg System

We present a global solution for an optimal feedback controller of the underactuated Heisenberg system or nonholonomic integrator. Employing a recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory, we circumvent a singularity caused by under actuation to develop a nonlinear optimal feedback control in an implicitly analytical form. The systematic procedure to deal with under actuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems.